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Betti Number Behavior for Nilpotent Lie Algebras

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Abstract

We discuss three general problems concerning the cohomology of a (real or complex) nilpotent Lie algebra: first of all, determining the Betti numbers exactly; second, determining the distribution these Betti numbers follow; and finally, estimating the size of the individual cohomology spaces or the total cohomology space. We show how spectral sequence arguments can contribute to a solution in a concrete setting. For one-dimensional extensions of a Heisenberg algebra, we determine the Betti numbers exactly. We then show that some families in this class have a M-shaped Betti number distribution, and construct the first examples with an even more exotic Betti number distribution. Finally, we discuss the construction of (co)homology classes for split metabelian Lie algebras, thus proving the Toral Rank Conjecture for this class of algebras.

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References

  1. Armstrong G. (1997). Unimodal Betti numbers for a class of nilpotent Lie algebras. Comm. Algebra 25(6): 1893–1915

    MATH  MathSciNet  Google Scholar 

  2. Armstrong G., Cairns G. and Jessup B. (1997). Explicit Betti numbers for a family of nilpotent Lie algebras, Proc. Amer. Math. Soc. 125(2): 381–385

    Article  MATH  MathSciNet  Google Scholar 

  3. Armstrong G. and Sigg S. (1996). On the cohomology of a particular class of nilpotent Lie algebras. Bull. Austral. Math. Soc. 54: 517–527

    Article  MATH  MathSciNet  Google Scholar 

  4. Cairns G. and Jessup B. (1997). New bounds on the Betti numbers of nilpotent Lie algebras. Comm. Algebra 25: 415–430

    MATH  MathSciNet  Google Scholar 

  5. Deninger C. and Singhof W. (1988). On the cohomology of nilpotent Lie algebras. Bull. Soc. Math. France 116: 3–14

    MATH  MathSciNet  Google Scholar 

  6. Dixmier J. (1955). Cohomologie des algèbres de Lie nilpotentes. Acta Sci. Math. 16: 246–250

    MATH  MathSciNet  Google Scholar 

  7. Halperin S.: Rational homotopy and torus action, In: Aspects of Topology. London Math. Soc. Lecture Notes, Ser. 93, Cambridge University Press, Cambridge, 1985, pp. 293–306.

  8. Pouseele H. On the cohomology of extensions by a Heisenberg Lie algebra, to appear in: Bull. Austral. Math. Soc.

  9. Pouseele, H. and Tirao, P.: Constructing Lie algebra homology, to appear in: J. Algebra.

  10. Santharoubane L.J. (1983). Cohomology of Heisenberg Lie algebras. Proc. Amer. Math. Soc. 87(1): 23–28

    Article  MATH  MathSciNet  Google Scholar 

  11. Tirao P. (2000). A refinement of the Toral Rank Conjecture for 2-step nilpotent Lie algebras. Proc. Amer. Math. Soc. 185(10): 2875–2878

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Katholieke Universiteit Leuven Campus Kortrijk, Etienne Sabbelaan 53, Kortrijk, 8500, Belgium

    Hannes Pouseele

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  1. Hannes Pouseele
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Correspondence to Hannes Pouseele.

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Pouseele, H. Betti Number Behavior for Nilpotent Lie Algebras. Geom Dedicata 122, 77–88 (2006). https://doi.org/10.1007/s10711-005-9029-9

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  • DOI: https://doi.org/10.1007/s10711-005-9029-9

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